added Typed-wf-deadend
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wadler
parent
9d5fed84a1
commit
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2 changed files with 866 additions and 1 deletions
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@ -623,7 +623,11 @@ In that case we have:
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Here, since `N` is well-typed, none of it's bound variables collide
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with `Γ`, and hence cannot collide with any free variable of `M`.
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*But* we can't make a similar guarantee for the *bound* variables
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of `M`, so substitution may break the invariants. Here is an example:
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of `M`, so substitution may break the invariants. Here are examples:
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(`λ "x" `→ `λ "y" `→ ` "x") (`λ "y" `→ ` "y")
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⟶
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(`λ "y" → (`λ "y" `→ ` "y"))
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ε , "z" `: `ℕ ⊢ (`λ "x" `→ `λ "y" → ` "x" · ` "y" · ` "z") (`λ "y" `→ ` "y" · ` "z")
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⟶
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@ -639,6 +643,26 @@ variable in `N` to not appear in `Γ`. Not clear how to maintain
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such a condition without the invariant, so I don't know how
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the proof works. Bugger!
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Consider a term with free variables, where every bound
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variable of the term is distinct from any free variable.
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(This is trivially true for a closed term.) Question: if
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I never reduce under lambda, do I ever need
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to perform renaming?
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It's easy to come up with a counter-example if I allow
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reduction under lambda.
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(λ y → (λ x → λ y → x y) y) ⟶ (λ y → (λ y′ → y y′))
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The above requires renaming. But if I remove the outer lambda
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(λ x → λ y → x y) y ⟶ (λ y → (λ y′ → y y′))
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then the term on the left violates the condition on free
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variables, and any term I can think of that causes problems
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also violates the condition. So I may be able to do something
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here.
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\begin{code}
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{-
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841
src/extra/Typed-wf-deadend.lagda
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841
src/extra/Typed-wf-deadend.lagda
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@ -0,0 +1,841 @@
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---
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title : "Typed: Typed Lambda term representation"
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layout : page
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permalink : /Typed
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---
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## Imports
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\begin{code}
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module Typed where
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\end{code}
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\begin{code}
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import Relation.Binary.PropositionalEquality as Eq
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open Eq using (_≡_; refl; sym; trans; cong; cong₂; _≢_)
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open import Data.Empty using (⊥; ⊥-elim)
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open import Data.List using (List; []; _∷_; _++_; map; foldr; filter)
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open import Data.Nat using (ℕ; zero; suc; _+_)
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open import Data.String using (String; _≟_)
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open import Data.Product using (_×_; proj₁; proj₂) renaming (_,_ to ⟨_,_⟩)
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open import Data.Sum using (_⊎_; inj₁; inj₂)
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open import Function using (_∘_)
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open import Relation.Nullary using (¬_; Dec; yes; no)
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open import Relation.Nullary.Negation using (¬?)
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open import Collections
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pattern [_] x = x ∷ []
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pattern [_,_] x y = x ∷ y ∷ []
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pattern [_,_,_] x y z = x ∷ y ∷ z ∷ []
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\end{code}
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## Syntax
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\begin{code}
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infix 4 _wf
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infix 4 _∉_
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infix 4 _∋_`:_
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infix 4 _⊢_`:_
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infixl 5 _,_`:_
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infixr 6 _`→_
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infix 6 `λ_`→_
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infixl 7 `if0_then_else_
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infix 8 `suc_ `pred_ `Y_
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infixl 9 _·_
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infix 10 S_
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Id : Set
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Id = String
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data Type : Set where
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`ℕ : Type
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_`→_ : Type → Type → Type
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data Env : Set where
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ε : Env
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_,_`:_ : Env → Id → Type → Env
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data Term : Set where
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`_ : Id → Term
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`λ_`→_ : Id → Term → Term
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_·_ : Term → Term → Term
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`zero : Term
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`suc_ : Term → Term
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`pred_ : Term → Term
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`if0_then_else_ : Term → Term → Term → Term
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`Y_ : Term → Term
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data _∋_`:_ : Env → Id → Type → Set where
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Z : ∀ {Γ A x}
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--------------------
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→ Γ , x `: A ∋ x `: A
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S_ : ∀ {Γ A B x w}
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→ Γ ∋ w `: B
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--------------------
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→ Γ , x `: A ∋ w `: B
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_∉_ : Id → Env → Set
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x ∉ Γ = ∀ {A} → ¬ (Γ ∋ x `: A)
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data _⊢_`:_ : Env → Term → Type → Set where
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Ax : ∀ {Γ A x}
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→ Γ ∋ x `: A
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--------------
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→ Γ ⊢ ` x `: A
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⊢λ : ∀ {Γ x N A B}
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→ x ∉ Γ
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→ Γ , x `: A ⊢ N `: B
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--------------------------
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→ Γ ⊢ (`λ x `→ N) `: A `→ B
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_·_ : ∀ {Γ L M A B}
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→ Γ ⊢ L `: A `→ B
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→ Γ ⊢ M `: A
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----------------
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→ Γ ⊢ L · M `: B
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⊢zero : ∀ {Γ}
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----------------
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→ Γ ⊢ `zero `: `ℕ
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⊢suc : ∀ {Γ M}
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→ Γ ⊢ M `: `ℕ
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-----------------
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→ Γ ⊢ `suc M `: `ℕ
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⊢pred : ∀ {Γ M}
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→ Γ ⊢ M `: `ℕ
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------------------
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→ Γ ⊢ `pred M `: `ℕ
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⊢if0 : ∀ {Γ L M N A}
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→ Γ ⊢ L `: `ℕ
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→ Γ ⊢ M `: A
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→ Γ ⊢ N `: A
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------------------------------
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→ Γ ⊢ `if0 L then M else N `: A
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⊢Y : ∀ {Γ M A}
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→ Γ ⊢ M `: A `→ A
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----------------
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→ Γ ⊢ `Y M `: A
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data _wf : Env → Set where
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empty :
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-----
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ε wf
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extend : ∀ {Γ x A}
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→ Γ wf
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→ x ∉ Γ
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-------------------------
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→ (Γ , x `: A) wf
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\end{code}
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## Test examples
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\begin{code}
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two : Term
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two = `suc `suc `zero
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⊢two : ε ⊢ two `: `ℕ
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⊢two = (⊢suc (⊢suc ⊢zero))
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plus : Term
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plus = `Y (`λ "p" `→ `λ "m" `→ `λ "n" `→ `if0 ` "m" then ` "n" else ` "p" · (`pred ` "m") · ` "n")
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⊢plus : ε ⊢ plus `: `ℕ `→ `ℕ `→ `ℕ
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⊢plus = (⊢Y (⊢λ p∉ (⊢λ m∉ (⊢λ n∉
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(⊢if0 (Ax ⊢m) (Ax ⊢n) (Ax ⊢p · (⊢pred (Ax ⊢m)) · Ax ⊢n))))))
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where
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⊢p = S S Z
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⊢m = S Z
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⊢n = Z
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Γ₀ = ε
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Γ₁ = Γ₀ , "p" `: `ℕ `→ `ℕ `→ `ℕ
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Γ₂ = Γ₁ , "m" `: `ℕ
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p∉ : "p" ∉ Γ₀
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p∉ ()
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m∉ : "m" ∉ Γ₁
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m∉ (S ())
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n∉ : "n" ∉ Γ₂
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n∉ (S S ())
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four : Term
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four = plus · two · two
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⊢four : ε ⊢ four `: `ℕ
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⊢four = ⊢plus · ⊢two · ⊢two
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Ch : Type
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Ch = (`ℕ `→ `ℕ) `→ `ℕ `→ `ℕ
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twoCh : Term
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twoCh = `λ "s" `→ `λ "z" `→ (` "s" · (` "s" · ` "z"))
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⊢twoCh : ε ⊢ twoCh `: Ch
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⊢twoCh = (⊢λ s∉ (⊢λ z∉ (Ax ⊢s · (Ax ⊢s · Ax ⊢z))))
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where
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⊢s = S Z
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⊢z = Z
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Γ₀ = ε
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Γ₁ = Γ₀ , "s" `: `ℕ `→ `ℕ
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s∉ : "s" ∉ ε
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s∉ ()
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z∉ : "z" ∉ Γ₁
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z∉ (S ())
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plusCh : Term
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plusCh = `λ "m" `→ `λ "n" `→ `λ "s" `→ `λ "z" `→
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` "m" · ` "s" · (` "n" · ` "s" · ` "z")
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⊢plusCh : ε ⊢ plusCh `: Ch `→ Ch `→ Ch
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⊢plusCh = (⊢λ m∉ (⊢λ n∉ (⊢λ s∉ (⊢λ z∉ (Ax ⊢m · Ax ⊢s · (Ax ⊢n · Ax ⊢s · Ax ⊢z))))))
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where
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⊢m = S S S Z
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⊢n = S S Z
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⊢s = S Z
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⊢z = Z
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Γ₀ = ε
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Γ₁ = Γ₀ , "m" `: Ch
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Γ₂ = Γ₁ , "n" `: Ch
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Γ₃ = Γ₂ , "s" `: `ℕ `→ `ℕ
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m∉ : "m" ∉ Γ₀
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m∉ ()
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n∉ : "n" ∉ Γ₁
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n∉ (S ())
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s∉ : "s" ∉ Γ₂
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s∉ (S S ())
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z∉ : "z" ∉ Γ₃
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z∉ (S S S ())
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fromCh : Term
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fromCh = `λ "m" `→ ` "m" · (`λ "s" `→ `suc ` "s") · `zero
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⊢fromCh : ε ⊢ fromCh `: Ch `→ `ℕ
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⊢fromCh = (⊢λ m∉ (Ax ⊢m · (⊢λ s∉ (⊢suc (Ax ⊢s))) · ⊢zero))
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where
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⊢m = Z
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⊢s = Z
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Γ₀ = ε
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Γ₁ = Γ₀ , "m" `: Ch
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m∉ : "m" ∉ Γ₀
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m∉ ()
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s∉ : "s" ∉ Γ₁
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s∉ (S ())
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fourCh : Term
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fourCh = fromCh · (plusCh · twoCh · twoCh)
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⊢fourCh : ε ⊢ fourCh `: `ℕ
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⊢fourCh = ⊢fromCh · (⊢plusCh · ⊢twoCh · ⊢twoCh)
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\end{code}
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## Erasure
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\begin{code}
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lookup : ∀ {Γ x A} → Γ ∋ x `: A → Id
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lookup {Γ , x `: A} Z = x
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lookup {Γ , x `: A} (S ⊢w) = lookup {Γ} ⊢w
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erase : ∀ {Γ M A} → Γ ⊢ M `: A → Term
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erase (Ax ⊢w) = ` lookup ⊢w
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erase (⊢λ {x = x} x∉ ⊢N) = `λ x `→ erase ⊢N
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erase (⊢L · ⊢M) = erase ⊢L · erase ⊢M
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erase (⊢zero) = `zero
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erase (⊢suc ⊢M) = `suc (erase ⊢M)
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erase (⊢pred ⊢M) = `pred (erase ⊢M)
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erase (⊢if0 ⊢L ⊢M ⊢N) = `if0 (erase ⊢L) then (erase ⊢M) else (erase ⊢N)
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erase (⊢Y ⊢M) = `Y (erase ⊢M)
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\end{code}
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### Properties of erasure
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\begin{code}
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cong₃ : ∀ {A B C D : Set} (f : A → B → C → D) {s t u v x y} →
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s ≡ t → u ≡ v → x ≡ y → f s u x ≡ f t v y
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cong₃ f refl refl refl = refl
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lookup-lemma : ∀ {Γ x A} → (⊢x : Γ ∋ x `: A) → lookup ⊢x ≡ x
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lookup-lemma Z = refl
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lookup-lemma (S ⊢w) = lookup-lemma ⊢w
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erase-lemma : ∀ {Γ M A} → (⊢M : Γ ⊢ M `: A) → erase ⊢M ≡ M
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erase-lemma (Ax ⊢x) = cong `_ (lookup-lemma ⊢x)
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erase-lemma (⊢λ {x = x} x∉ ⊢N) = cong (`λ x `→_) (erase-lemma ⊢N)
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erase-lemma (⊢L · ⊢M) = cong₂ _·_ (erase-lemma ⊢L) (erase-lemma ⊢M)
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erase-lemma (⊢zero) = refl
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erase-lemma (⊢suc ⊢M) = cong `suc_ (erase-lemma ⊢M)
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erase-lemma (⊢pred ⊢M) = cong `pred_ (erase-lemma ⊢M)
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erase-lemma (⊢if0 ⊢L ⊢M ⊢N) = cong₃ `if0_then_else_
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(erase-lemma ⊢L) (erase-lemma ⊢M) (erase-lemma ⊢N)
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erase-lemma (⊢Y ⊢M) = cong `Y_ (erase-lemma ⊢M)
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\end{code}
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## Substitution
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### Lists as sets
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\begin{code}
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open Collections.CollectionDec (Id) (_≟_)
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\end{code}
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### Free variables
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\begin{code}
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free : Term → List Id
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free (` x) = [ x ]
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free (`λ x `→ N) = free N \\ x
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free (L · M) = free L ++ free M
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free (`zero) = []
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free (`suc M) = free M
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free (`pred M) = free M
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free (`if0 L then M else N) = free L ++ free M ++ free N
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free (`Y M) = free M
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\end{code}
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### Identifier maps
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\begin{code}
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∅ : Id → Term
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∅ x = ` x
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infixl 5 _,_↦_
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_,_↦_ : (Id → Term) → Id → Term → (Id → Term)
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(ρ , x ↦ M) w with w ≟ x
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... | yes _ = M
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... | no _ = ρ w
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\end{code}
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### Substitution
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\begin{code}
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subst : (Id → Term) → Term → Term
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subst ρ (` x) = ρ x
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subst ρ (`λ x `→ N) = `λ x `→ subst (ρ , x ↦ ` x) N
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subst ρ (L · M) = subst ρ L · subst ρ M
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subst ρ (`zero) = `zero
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subst ρ (`suc M) = `suc (subst ρ M)
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subst ρ (`pred M) = `pred (subst ρ M)
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subst ρ (`if0 L then M else N)
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= `if0 (subst ρ L) then (subst ρ M) else (subst ρ N)
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subst ρ (`Y M) = `Y (subst ρ M)
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_[_:=_] : Term → Id → Term → Term
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N [ x := M ] = subst (∅ , x ↦ M) N
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\end{code}
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### Testing substitution
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\begin{code}
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_ : (` "s" · ` "s" · ` "z") [ "z" := `zero ] ≡ (` "s" · ` "s" · `zero)
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_ = refl
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_ : (` "s" · ` "s" · ` "z") [ "s" := (`λ "m" `→ `suc ` "m") ] [ "z" := `zero ]
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≡ (`λ "m" `→ `suc ` "m") · (`λ "m" `→ `suc ` "m") · `zero
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_ = refl
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_ : (`λ "m" `→ ` "m" · ` "n") [ "n" := ` "p" · ` "q" ]
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≡ `λ "m" `→ ` "m" · (` "p" · ` "q")
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_ = refl
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_ : subst (∅ , "m" ↦ ` "p" , "n" ↦ ` "q") (` "m" · ` "n") ≡ (` "p" · ` "q")
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_ = refl
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\end{code}
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## Values
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\begin{code}
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data Value : Term → Set where
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Zero :
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----------
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Value `zero
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Suc : ∀ {V}
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→ Value V
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--------------
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→ Value (`suc V)
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Fun : ∀ {x N}
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---------------
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→ Value (`λ x `→ N)
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\end{code}
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## Reduction
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\begin{code}
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infix 4 _⟶_
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data _⟶_ : Term → Term → Set where
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ξ-·₁ : ∀ {L L′ M}
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→ L ⟶ L′
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-----------------
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→ L · M ⟶ L′ · M
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ξ-·₂ : ∀ {V M M′}
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→ Value V
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→ M ⟶ M′
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-----------------
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→ V · M ⟶ V · M′
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β-`→ : ∀ {x N V}
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→ Value V
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---------------------------------
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→ (`λ x `→ N) · V ⟶ N [ x := V ]
|
||||
|
||||
ξ-suc : ∀ {M M′}
|
||||
→ M ⟶ M′
|
||||
-------------------
|
||||
→ `suc M ⟶ `suc M′
|
||||
|
||||
ξ-pred : ∀ {M M′}
|
||||
→ M ⟶ M′
|
||||
---------------------
|
||||
→ `pred M ⟶ `pred M′
|
||||
|
||||
β-pred-zero :
|
||||
----------------------
|
||||
`pred `zero ⟶ `zero
|
||||
|
||||
β-pred-suc : ∀ {V}
|
||||
→ Value V
|
||||
---------------------
|
||||
→ `pred (`suc V) ⟶ V
|
||||
|
||||
ξ-if0 : ∀ {L L′ M N}
|
||||
→ L ⟶ L′
|
||||
-----------------------------------------------
|
||||
→ `if0 L then M else N ⟶ `if0 L′ then M else N
|
||||
|
||||
β-if0-zero : ∀ {M N}
|
||||
-------------------------------
|
||||
→ `if0 `zero then M else N ⟶ M
|
||||
|
||||
β-if0-suc : ∀ {V M N}
|
||||
→ Value V
|
||||
----------------------------------
|
||||
→ `if0 (`suc V) then M else N ⟶ N
|
||||
|
||||
ξ-Y : ∀ {M M′}
|
||||
→ M ⟶ M′
|
||||
---------------
|
||||
→ `Y M ⟶ `Y M′
|
||||
|
||||
β-Y : ∀ {V x N}
|
||||
→ Value V
|
||||
→ V ≡ `λ x `→ N
|
||||
-------------------------
|
||||
→ `Y V ⟶ N [ x := `Y V ]
|
||||
\end{code}
|
||||
|
||||
## Reflexive and transitive closure
|
||||
|
||||
\begin{code}
|
||||
infix 2 _⟶*_
|
||||
infix 1 begin_
|
||||
infixr 2 _⟶⟨_⟩_
|
||||
infix 3 _∎
|
||||
|
||||
data _⟶*_ : Term → Term → Set where
|
||||
|
||||
_∎ : ∀ (M : Term)
|
||||
-------------
|
||||
→ M ⟶* M
|
||||
|
||||
_⟶⟨_⟩_ : ∀ (L : Term) {M N}
|
||||
→ L ⟶ M
|
||||
→ M ⟶* N
|
||||
---------
|
||||
→ L ⟶* N
|
||||
|
||||
begin_ : ∀ {M N} → (M ⟶* N) → (M ⟶* N)
|
||||
begin M⟶*N = M⟶*N
|
||||
\end{code}
|
||||
|
||||
## Canonical forms
|
||||
|
||||
\begin{code}
|
||||
data Canonical : Term → Type → Set where
|
||||
|
||||
Zero :
|
||||
-------------------
|
||||
Canonical `zero `ℕ
|
||||
|
||||
Suc : ∀ {V}
|
||||
→ Canonical V `ℕ
|
||||
----------------------
|
||||
→ Canonical (`suc V) `ℕ
|
||||
|
||||
Fun : ∀ {x N A B}
|
||||
→ ε , x `: A ⊢ N `: B
|
||||
-------------------------------
|
||||
→ Canonical (`λ x `→ N) (A `→ B)
|
||||
\end{code}
|
||||
|
||||
## Canonical forms lemma
|
||||
|
||||
Every typed value is canonical.
|
||||
|
||||
\begin{code}
|
||||
canonical : ∀ {V A}
|
||||
→ ε ⊢ V `: A
|
||||
→ Value V
|
||||
-------------
|
||||
→ Canonical V A
|
||||
canonical ⊢zero Zero = Zero
|
||||
canonical (⊢suc ⊢V) (Suc VV) = Suc (canonical ⊢V VV)
|
||||
canonical (⊢λ x∉ ⊢N) Fun = Fun ⊢N
|
||||
\end{code}
|
||||
|
||||
Every canonical form has a type and a value.
|
||||
|
||||
\begin{code}
|
||||
type : ∀ {V A}
|
||||
→ Canonical V A
|
||||
--------------
|
||||
→ ε ⊢ V `: A
|
||||
type Zero = ⊢zero
|
||||
type (Suc CV) = ⊢suc (type CV)
|
||||
type (Fun {x = x} ⊢N) = ⊢λ x∉ ⊢N
|
||||
where
|
||||
x∉ : x ∉ ε
|
||||
x∉ ()
|
||||
|
||||
value : ∀ {V A}
|
||||
→ Canonical V A
|
||||
-------------
|
||||
→ Value V
|
||||
value Zero = Zero
|
||||
value (Suc CV) = Suc (value CV)
|
||||
value (Fun ⊢N) = Fun
|
||||
\end{code}
|
||||
|
||||
## Progress
|
||||
|
||||
\begin{code}
|
||||
data Progress (M : Term) (A : Type) : Set where
|
||||
step : ∀ {N}
|
||||
→ M ⟶ N
|
||||
----------
|
||||
→ Progress M A
|
||||
done :
|
||||
Canonical M A
|
||||
-------------
|
||||
→ Progress M A
|
||||
|
||||
progress : ∀ {M A} → ε ⊢ M `: A → Progress M A
|
||||
progress (Ax ())
|
||||
progress (⊢λ x∉ ⊢N) = done (Fun ⊢N)
|
||||
progress (⊢L · ⊢M) with progress ⊢L
|
||||
... | step L⟶L′ = step (ξ-·₁ L⟶L′)
|
||||
... | done (Fun _) with progress ⊢M
|
||||
... | step M⟶M′ = step (ξ-·₂ Fun M⟶M′)
|
||||
... | done CM = step (β-`→ (value CM))
|
||||
progress ⊢zero = done Zero
|
||||
progress (⊢suc ⊢M) with progress ⊢M
|
||||
... | step M⟶M′ = step (ξ-suc M⟶M′)
|
||||
... | done CM = done (Suc CM)
|
||||
progress (⊢pred ⊢M) with progress ⊢M
|
||||
... | step M⟶M′ = step (ξ-pred M⟶M′)
|
||||
... | done Zero = step β-pred-zero
|
||||
... | done (Suc CM) = step (β-pred-suc (value CM))
|
||||
progress (⊢if0 ⊢L ⊢M ⊢N) with progress ⊢L
|
||||
... | step L⟶L′ = step (ξ-if0 L⟶L′)
|
||||
... | done Zero = step β-if0-zero
|
||||
... | done (Suc CM) = step (β-if0-suc (value CM))
|
||||
progress (⊢Y ⊢M) with progress ⊢M
|
||||
... | step M⟶M′ = step (ξ-Y M⟶M′)
|
||||
... | done (Fun _) = step (β-Y Fun refl)
|
||||
\end{code}
|
||||
|
||||
|
||||
## Preservation
|
||||
|
||||
### Domain of an environment
|
||||
|
||||
\begin{code}
|
||||
{-
|
||||
dom : Env → List Id
|
||||
dom ε = []
|
||||
dom (Γ , x `: A) = x ∷ dom Γ
|
||||
|
||||
dom-lemma : ∀ {Γ y B} → Γ ∋ y `: B → y ∈ dom Γ
|
||||
dom-lemma Z = here
|
||||
dom-lemma (S x≢y ⊢y) = there (dom-lemma ⊢y)
|
||||
|
||||
free-lemma : ∀ {Γ M A} → Γ ⊢ M `: A → free M ⊆ dom Γ
|
||||
free-lemma (Ax ⊢x) w∈ with w∈
|
||||
... | here = dom-lemma ⊢x
|
||||
... | there ()
|
||||
free-lemma {Γ} (⊢λ {N = N} ⊢N) = ∷-to-\\ (free-lemma ⊢N)
|
||||
free-lemma (⊢L · ⊢M) w∈ with ++-to-⊎ w∈
|
||||
... | inj₁ ∈L = free-lemma ⊢L ∈L
|
||||
... | inj₂ ∈M = free-lemma ⊢M ∈M
|
||||
free-lemma ⊢zero ()
|
||||
free-lemma (⊢suc ⊢M) w∈ = free-lemma ⊢M w∈
|
||||
free-lemma (⊢pred ⊢M) w∈ = free-lemma ⊢M w∈
|
||||
free-lemma (⊢if0 ⊢L ⊢M ⊢N) w∈
|
||||
with ++-to-⊎ w∈
|
||||
... | inj₁ ∈L = free-lemma ⊢L ∈L
|
||||
... | inj₂ ∈MN with ++-to-⊎ ∈MN
|
||||
... | inj₁ ∈M = free-lemma ⊢M ∈M
|
||||
... | inj₂ ∈N = free-lemma ⊢N ∈N
|
||||
free-lemma (⊢Y ⊢M) w∈ = free-lemma ⊢M w∈
|
||||
-}
|
||||
\end{code}
|
||||
|
||||
### Renaming
|
||||
|
||||
Let's try an example. The result I want to prove is:
|
||||
|
||||
⊢subst : ∀ {Γ Δ ρ}
|
||||
→ (∀ {x A} → Γ ∋ x `: A → Δ ⊢ ρ x `: A)
|
||||
-----------------------------------------------
|
||||
→ (∀ {M A} → Γ ⊢ M `: A → Δ ⊢ subst ρ M `: A)
|
||||
|
||||
For this to work, I need to know that neither `Δ` or any of the
|
||||
bound variables in `ρ x` will collide with any bound variable in `M`.
|
||||
How can I establish this?
|
||||
|
||||
In particular, I need to check that the conditions for ordinary
|
||||
substitution are sufficient to establish the required invariants.
|
||||
In that case we have:
|
||||
|
||||
⊢substitution : ∀ {Γ x A N B M} →
|
||||
Γ , x `: A ⊢ N `: B →
|
||||
Γ ⊢ M `: A →
|
||||
--------------------
|
||||
Γ ⊢ N [ x := M ] `: B
|
||||
|
||||
Here, since `N` is well-typed, none of it's bound variables collide
|
||||
with `Γ`, and hence cannot collide with any free variable of `M`.
|
||||
*But* we can't make a similar guarantee for the *bound* variables
|
||||
of `M`, so substitution may break the invariants. Here are examples:
|
||||
|
||||
(`λ "x" `→ `λ "y" `→ ` "x") (`λ "y" `→ ` "y")
|
||||
⟶
|
||||
(`λ "y" → (`λ "y" `→ ` "y"))
|
||||
|
||||
ε , "z" `: `ℕ ⊢ (`λ "x" `→ `λ "y" → ` "x" · ` "y" · ` "z") (`λ "y" `→ ` "y" · ` "z")
|
||||
⟶
|
||||
ε , "z" `: `ℕ ⊢ (`λ "y" → (`λ "y" `→ ` "y" · ` "z") · ` "y" · ` "z")
|
||||
|
||||
This doesn't maintain the invariant, but doesn't break either.
|
||||
But I don't know how to prove it never breaks. Maybe I can come
|
||||
up with an example that does break after a few steps. Or, maybe
|
||||
I don't need the nested variables to be unique. Maybe all I need
|
||||
is for the free variables in each `ρ x` to be distinct from any
|
||||
of the bound variables in `N`. But this requires every bound
|
||||
variable in `N` to not appear in `Γ`. Not clear how to maintain
|
||||
such a condition without the invariant, so I don't know how
|
||||
the proof works. Bugger!
|
||||
|
||||
Consider a term with free variables, where every bound
|
||||
variable of the term is distinct from any free variable.
|
||||
(This is trivially true for a closed term.) Question: if
|
||||
I never reduce under lambda, do I ever need
|
||||
to perform renaming?
|
||||
|
||||
It's easy to come up with a counter-example if I allow
|
||||
reduction under lambda.
|
||||
|
||||
(λ y → (λ x → λ y → x y) y) ⟶ (λ y → (λ y′ → y y′))
|
||||
|
||||
The above requires renaming. But if I remove the outer lambda
|
||||
|
||||
(λ x → λ y → x y) y ⟶ (λ y → (λ y′ → y y′))
|
||||
|
||||
then the term on the left violates the condition on free
|
||||
variables, and any term I can think of that causes problems
|
||||
also violates the condition. So I may be able to do something
|
||||
here.
|
||||
|
||||
|
||||
\begin{code}
|
||||
{-
|
||||
⊢rename : ∀ {Γ Δ xs}
|
||||
→ (∀ {x A} → Γ ∋ x `: A → Δ ∋ x `: A)
|
||||
--------------------------------------------------
|
||||
→ (∀ {M A} → Γ ⊢ M `: A → Δ ⊢ M `: A)
|
||||
⊢rename ⊢σ (Ax ⊢x) = Ax (⊢σ ⊢x)
|
||||
⊢rename {Γ} {Δ} ⊢σ (⊢λ {x = x} {N = N} {A = A} x∉Γ ⊢N)
|
||||
= ⊢λ x∉Δ (⊢rename {Γ′} {Δ′} ⊢σ′ ⊢N)
|
||||
where
|
||||
Γ′ = Γ , x `: A
|
||||
Δ′ = Δ , x `: A
|
||||
xs′ = x ∷ xs
|
||||
|
||||
⊢σ′ : ∀ {w B} → w ∈ xs′ → Γ′ ∋ w `: B → Δ′ ∋ w `: B
|
||||
⊢σ′ w∈′ Z = Z
|
||||
⊢σ′ w∈′ (S w≢ ⊢w) = S w≢ (⊢σ ∈w ⊢w)
|
||||
where
|
||||
∈w = there⁻¹ w∈′ w≢
|
||||
|
||||
⊆xs′ : free N ⊆ xs′
|
||||
⊆xs′ = \\-to-∷ ⊆xs
|
||||
⊢rename ⊢σ ⊆xs (⊢L · ⊢M) = ⊢rename ⊢σ L⊆ ⊢L · ⊢rename ⊢σ M⊆ ⊢M
|
||||
where
|
||||
L⊆ = trans-⊆ ⊆-++₁ ⊆xs
|
||||
M⊆ = trans-⊆ ⊆-++₂ ⊆xs
|
||||
⊢rename ⊢σ ⊆xs (⊢zero) = ⊢zero
|
||||
⊢rename ⊢σ ⊆xs (⊢suc ⊢M) = ⊢suc (⊢rename ⊢σ ⊆xs ⊢M)
|
||||
⊢rename ⊢σ ⊆xs (⊢pred ⊢M) = ⊢pred (⊢rename ⊢σ ⊆xs ⊢M)
|
||||
⊢rename ⊢σ ⊆xs (⊢if0 {L = L} ⊢L ⊢M ⊢N)
|
||||
= ⊢if0 (⊢rename ⊢σ L⊆ ⊢L) (⊢rename ⊢σ M⊆ ⊢M) (⊢rename ⊢σ N⊆ ⊢N)
|
||||
where
|
||||
L⊆ = trans-⊆ ⊆-++₁ ⊆xs
|
||||
M⊆ = trans-⊆ ⊆-++₁ (trans-⊆ (⊆-++₂ {free L}) ⊆xs)
|
||||
N⊆ = trans-⊆ ⊆-++₂ (trans-⊆ (⊆-++₂ {free L}) ⊆xs)
|
||||
⊢rename ⊢σ ⊆xs (⊢Y ⊢M) = ⊢Y (⊢rename ⊢σ ⊆xs ⊢M)
|
||||
-}
|
||||
\end{code}
|
||||
|
||||
|
||||
### Substitution preserves types
|
||||
|
||||
\begin{code}
|
||||
{-
|
||||
⊢subst : ∀ {Γ Δ xs ys ρ}
|
||||
→ (∀ {x} → x ∈ xs → free (ρ x) ⊆ ys)
|
||||
→ (∀ {x A} → x ∈ xs → Γ ∋ x `: A → Δ ⊢ ρ x `: A)
|
||||
-------------------------------------------------------------
|
||||
→ (∀ {M A} → free M ⊆ xs → Γ ⊢ M `: A → Δ ⊢ subst ys ρ M `: A)
|
||||
⊢subst Σ ⊢ρ ⊆xs (Ax ⊢x)
|
||||
= ⊢ρ (⊆xs here) ⊢x
|
||||
⊢subst {Γ} {Δ} {xs} {ys} {ρ} Σ ⊢ρ ⊆xs (⊢λ {x = x} {N = N} {A = A} ⊢N)
|
||||
= ⊢λ {x = y} {A = A} (⊢subst {Γ′} {Δ′} {xs′} {ys′} {ρ′} Σ′ ⊢ρ′ ⊆xs′ ⊢N)
|
||||
where
|
||||
y = fresh ys
|
||||
Γ′ = Γ , x `: A
|
||||
Δ′ = Δ , y `: A
|
||||
xs′ = x ∷ xs
|
||||
ys′ = y ∷ ys
|
||||
ρ′ = ρ , x ↦ ` y
|
||||
|
||||
Σ′ : ∀ {w} → w ∈ xs′ → free (ρ′ w) ⊆ ys′
|
||||
Σ′ {w} w∈′ with w ≟ x
|
||||
... | yes refl = ⊆-++₁
|
||||
... | no w≢ = ⊆-++₂ ∘ Σ (there⁻¹ w∈′ w≢)
|
||||
|
||||
⊆xs′ : free N ⊆ xs′
|
||||
⊆xs′ = \\-to-∷ ⊆xs
|
||||
|
||||
⊢σ : ∀ {w C} → w ∈ ys → Δ ∋ w `: C → Δ′ ∋ w `: C
|
||||
⊢σ w∈ ⊢w = S (fresh-lemma w∈) ⊢w
|
||||
|
||||
⊢ρ′ : ∀ {w C} → w ∈ xs′ → Γ′ ∋ w `: C → Δ′ ⊢ ρ′ w `: C
|
||||
⊢ρ′ {w} _ Z with w ≟ x
|
||||
... | yes _ = Ax Z
|
||||
... | no w≢ = ⊥-elim (w≢ refl)
|
||||
⊢ρ′ {w} w∈′ (S w≢ ⊢w) with w ≟ x
|
||||
... | yes refl = ⊥-elim (w≢ refl)
|
||||
... | no _ = ⊢rename {Δ} {Δ′} {ys} ⊢σ (Σ w∈) (⊢ρ w∈ ⊢w)
|
||||
where
|
||||
w∈ = there⁻¹ w∈′ w≢
|
||||
|
||||
⊢subst Σ ⊢ρ ⊆xs (⊢L · ⊢M)
|
||||
= ⊢subst Σ ⊢ρ L⊆ ⊢L · ⊢subst Σ ⊢ρ M⊆ ⊢M
|
||||
where
|
||||
L⊆ = trans-⊆ ⊆-++₁ ⊆xs
|
||||
M⊆ = trans-⊆ ⊆-++₂ ⊆xs
|
||||
⊢subst Σ ⊢ρ ⊆xs ⊢zero = ⊢zero
|
||||
⊢subst Σ ⊢ρ ⊆xs (⊢suc ⊢M) = ⊢suc (⊢subst Σ ⊢ρ ⊆xs ⊢M)
|
||||
⊢subst Σ ⊢ρ ⊆xs (⊢pred ⊢M) = ⊢pred (⊢subst Σ ⊢ρ ⊆xs ⊢M)
|
||||
⊢subst Σ ⊢ρ ⊆xs (⊢if0 {L = L} ⊢L ⊢M ⊢N)
|
||||
= ⊢if0 (⊢subst Σ ⊢ρ L⊆ ⊢L) (⊢subst Σ ⊢ρ M⊆ ⊢M) (⊢subst Σ ⊢ρ N⊆ ⊢N)
|
||||
where
|
||||
L⊆ = trans-⊆ ⊆-++₁ ⊆xs
|
||||
M⊆ = trans-⊆ ⊆-++₁ (trans-⊆ (⊆-++₂ {free L}) ⊆xs)
|
||||
N⊆ = trans-⊆ ⊆-++₂ (trans-⊆ (⊆-++₂ {free L}) ⊆xs)
|
||||
⊢subst Σ ⊢ρ ⊆xs (⊢Y ⊢M) = ⊢Y (⊢subst Σ ⊢ρ ⊆xs ⊢M)
|
||||
|
||||
⊢substitution : ∀ {Γ x A N B M} →
|
||||
Γ , x `: A ⊢ N `: B →
|
||||
Γ ⊢ M `: A →
|
||||
--------------------
|
||||
Γ ⊢ N [ x := M ] `: B
|
||||
⊢substitution {Γ} {x} {A} {N} {B} {M} ⊢N ⊢M =
|
||||
⊢subst {Γ′} {Γ} {xs} {ys} {ρ} Σ ⊢ρ {N} {B} ⊆xs ⊢N
|
||||
where
|
||||
Γ′ = Γ , x `: A
|
||||
xs = free N
|
||||
ys = free M ++ (free N \\ x)
|
||||
ρ = ∅ , x ↦ M
|
||||
|
||||
Σ : ∀ {w} → w ∈ xs → free (ρ w) ⊆ ys
|
||||
Σ {w} w∈ y∈ with w ≟ x
|
||||
... | yes _ = ⊆-++₁ y∈
|
||||
... | no w≢ rewrite ∈-[_] y∈ = ⊆-++₂ (∈-≢-to-\\ w∈ w≢)
|
||||
|
||||
⊢ρ : ∀ {w B} → w ∈ xs → Γ′ ∋ w `: B → Γ ⊢ ρ w `: B
|
||||
⊢ρ {w} w∈ Z with w ≟ x
|
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... | yes _ = ⊢M
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... | no w≢ = ⊥-elim (w≢ refl)
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⊢ρ {w} w∈ (S w≢ ⊢w) with w ≟ x
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... | yes refl = ⊥-elim (w≢ refl)
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... | no _ = Ax ⊢w
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⊆xs : free N ⊆ xs
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⊆xs x∈ = x∈
|
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-}
|
||||
\end{code}
|
||||
|
||||
### Preservation
|
||||
|
||||
\begin{code}
|
||||
{-
|
||||
preservation : ∀ {Γ M N A}
|
||||
→ Γ ⊢ M `: A
|
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→ M ⟶ N
|
||||
---------
|
||||
→ Γ ⊢ N `: A
|
||||
preservation (Ax ⊢x) ()
|
||||
preservation (⊢λ ⊢N) ()
|
||||
preservation (⊢L · ⊢M) (ξ-·₁ L⟶) = preservation ⊢L L⟶ · ⊢M
|
||||
preservation (⊢V · ⊢M) (ξ-·₂ _ M⟶) = ⊢V · preservation ⊢M M⟶
|
||||
preservation ((⊢λ ⊢N) · ⊢W) (β-`→ _) = ⊢substitution ⊢N ⊢W
|
||||
preservation (⊢zero) ()
|
||||
preservation (⊢suc ⊢M) (ξ-suc M⟶) = ⊢suc (preservation ⊢M M⟶)
|
||||
preservation (⊢pred ⊢M) (ξ-pred M⟶) = ⊢pred (preservation ⊢M M⟶)
|
||||
preservation (⊢pred ⊢zero) (β-pred-zero) = ⊢zero
|
||||
preservation (⊢pred (⊢suc ⊢M)) (β-pred-suc _) = ⊢M
|
||||
preservation (⊢if0 ⊢L ⊢M ⊢N) (ξ-if0 L⟶) = ⊢if0 (preservation ⊢L L⟶) ⊢M ⊢N
|
||||
preservation (⊢if0 ⊢zero ⊢M ⊢N) β-if0-zero = ⊢M
|
||||
preservation (⊢if0 (⊢suc ⊢V) ⊢M ⊢N) (β-if0-suc _) = ⊢N
|
||||
preservation (⊢Y ⊢M) (ξ-Y M⟶) = ⊢Y (preservation ⊢M M⟶)
|
||||
preservation (⊢Y (⊢λ ⊢N)) (β-Y _ refl) = ⊢substitution ⊢N (⊢Y (⊢λ ⊢N))
|
||||
-}
|
||||
\end{code}
|
||||
|
||||
## Normalise
|
||||
|
||||
\begin{code}
|
||||
{-
|
||||
data Normalise {M A} (⊢M : ε ⊢ M `: A) : Set where
|
||||
out-of-gas : ∀ {N} → M ⟶* N → ε ⊢ N `: A → Normalise ⊢M
|
||||
normal : ∀ {V} → ℕ → Canonical V A → M ⟶* V → Normalise ⊢M
|
||||
|
||||
normalise : ∀ {L A} → ℕ → (⊢L : ε ⊢ L `: A) → Normalise ⊢L
|
||||
normalise {L} zero ⊢L = out-of-gas (L ∎) ⊢L
|
||||
normalise {L} (suc m) ⊢L with progress ⊢L
|
||||
... | done CL = normal (suc m) CL (L ∎)
|
||||
... | step L⟶M with preservation ⊢L L⟶M
|
||||
... | ⊢M with normalise m ⊢M
|
||||
... | out-of-gas M⟶*N ⊢N = out-of-gas (L ⟶⟨ L⟶M ⟩ M⟶*N) ⊢N
|
||||
... | normal n CV M⟶*V = normal n CV (L ⟶⟨ L⟶M ⟩ M⟶*V)
|
||||
-}
|
||||
\end{code}
|
||||
|
||||
Loading…
Reference in a new issue